For Milnor, statistical, and minimal attractors, we construct examples of smooth flows ϕ on S^2 for which the attractor of the Cartesian square of ϕ is smaller than the Cartesian square of the attractor of ϕ. In the example for the minimal attractors, the flow ϕ also has a global physical measure such that its square does not coincide with the global physical measure of the square of ϕ.
For a wide class of dynamical systems known as Pixton diffeomorphisms the topological conjugacy class is completely defined by the Hopf knot equivalence class, i.e. the knot whose equivalence class under homotopy of the loops is a generator of the fundamental group π1(S2×S1)π1(S2×S1). Moreover, any Hopf knot can be realized by a Pixton diffeomorphism. Nevertheless, the number of the classes of topological conjugacy of these diffeomorphisms is still unknown. This problem can be reduced to finding topological invariants of Hopf knots. In the present paper we describe a first order invariant for these knots. This result allows one to model countable families of pairwise non-equivalent Hopf knots and, therefore, infinite set of topologically non-conjugate Pixton diffeomorphisms.